Skill: Represent linear relationships using graphs

A linear relationship can be shown by plotting its ordered pairs $(x,y)$ on a coordinate plane. The independent quantity, $x$, goes on the horizontal axis, and the dependent quantity, $y$, goes on the vertical axis. The points will form a straight line.

Example

Graph the relationship

$$y=2x+1.$$

Step 1: Make a table of values

Choose a few values for $x$ and calculate $y$.

$$\begin{array}{c|c} x & y=2x+1\\ \hline -1 & 2(-1)+1=-1\\ 0 & 2(0)+1=1\\ 1 & 2(1)+1=3\\ 2 & 2(2)+1=5 \end{array}$$

This gives the ordered pairs

$$(-1,-1),\ (0,1),\ (1,3),\ (2,5).$$

Step 2: Label the axes

Draw a coordinate plane. Label the horizontal axis $x$ and the vertical axis $y$. Use an even scale, such as counting by ones.

Step 3: Plot the points

Plot each ordered pair:

• $(-1,-1)$: move left $1$ and down $1$.
• $(0,1)$: move up to $1$ on the $y$-axis.
• $(1,3)$: move right $1$ and up to $3$.
• $(2,5)$: move right $2$ and up to $5$.

Step 4: Draw the line

Use a ruler to draw a straight line through the points. The line represents all the ordered pairs that follow the relationship $y=2x+1$.

The number $2$ is the slope, so $y$ increases by $2$ whenever $x$ increases by $1$. The number $1$ is the $y$-intercept, so the line crosses the $y$-axis at $(0,1)$.

To check a graph of a linear relationship, make sure the points follow the equation and line up in a straight line.